Exact three- and four-state reconstruction thresholds and four-state tree capacity
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the exact reconstruction threshold \(d\lambda^2\gt 1\), with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Three other possible matches are listed below the card, among them Corollary 1.2(Exact weak-recovery threshold for three communities) in the same paper and Theorem 1.1 in A Capacity Criterion for Four-State Potts Reconstruction on Trees.
For every integer b\ge2 in the regular model and every real d>1 in the Poisson model, reconstruction occurs if and only if
d\lambda^2>1,
with d=b in the regular case. At every admissible equality d\lambda^2=1, there is non-reconstruction.
Three more possible matches
Corollary 1.2(Exact weak-recovery threshold for three communities)in The exact reconstruction threshold for the three-state symmetric channel, 25 September 2026.
Theorem 1.1in A Capacity Criterion for Four-State Potts Reconstruction on Trees, 5 October 2026.
Theorem 1.1in The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model, 5 October 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when \(d\lambda^2\gt 1\); we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival. The proof uses reproducible exact-arithmetic verification of polynomial inequalities.
For the ferromagnetic four-state broadcast model with \(0\lt \lambda\lt 1\), we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances \(\lambda^{-2|e|}\) is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.
We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when \(d\lambda^2\gt 1\), with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates \(a,b\gt 0\), and mean degree \((a+2b)/3\gt 1\): recovery is possible exactly when \((a-b)^2\gt 3(a+2b)\). Above this threshold it is achievable in \(O(n\log n)\) time; at or below it, weak recovery is information-theoretically impossible.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express a weaker statement than the advertised result (their term: “weaker-statement”).
Their note:Only the supercritical (Kesten-Stigum) direction for the three-state symmetric channel is stated: regular_supercritical (hcrit : 1 < b * lam^2) : Reconstructs.. and poisson_supercritical (hd : 1 < d) (hcrit : 1 < d * lam^2) : Reconstructs.. (advantage ∑' y, marginal y * (∑ i |posterior y i - 1/3|)/2 converges to L > 0), docs: 'Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them'. The summary’s headline is the EXACT threshold dλ²>1 with nonreconstruction at equality, plus the four-state ferromagnetic case and the SBM weak-recovery threshold: none of those is stated (the formalized direction is the classical one, the sharpness is missing).
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:(a) Non-reconstruction at and below dλ² = 1 (the ‘exact’ part of the threshold, and the new direction — reconstruction above Kesten–Stigum is classical) is not stated for either model; (b) every four-state ferromagnetic result (threshold on regular and Poisson trees, nonreconstruction at equality, and the L³-capacity criterion) is not stated; (c) the three-community SBM weak-recovery threshold is not stated. (a) for three states and (c) are declared; (a) for four states and (b) are not.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves the supercritical direction of reconstruction for the symmetric three-state broadcast channel, whose parameter satisfies $-1/2\le\lambda\le1$. On a regular $b$-ary tree, reconstruction holds when $b\lambda^2>1$; on an observed Poisson Galton–Watson tree of mean $d$, it holds when $d\lambda^2>1$. In each case the root-estimation advantage converges to a positive limit, with the Poisson advantage averaged over trees and spins.
The selected statements cover reconstruction above the Kesten–Stigum threshold. Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them.
AI-drafted comparisons, with no recorded human review
Supercritical reconstruction on regular and Poisson treesThreeStateSupercritical
Unreviewed AI draft: states part of a paper’s theorem. States the reconstruction direction of the exact-threshold theorem (reconstruction when dλ² > 1 on regular and Poisson trees). Non-reconstruction at or below the threshold, the other direction, is outside the statement.
Three-state reconstruction above the Kesten–Stigum thresholdThreeStateTreeClauses
Unreviewed AI draft: states part of a paper’s theorem. States the same reconstruction direction above the Kesten–Stigum threshold as ThreeStateSupercritical, in a separately written statement; non-reconstruction is outside it.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Supercritical reconstruction on regular and Poisson treesThreeStateSupercritical
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 229The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 229Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Investigate a calculation that did not decide the claim
Family 229An independent calculation did not decide the claim, or needs data absent from its source.Open the form
Family 229An independent calculation did not decide the claim, or needs data absent from its source.Open the form
Why this passage is shown
Theorem 1.1 (Exact threshold)
In The exact reconstruction threshold for the three-state symmetric channel, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 67 s; peak memory 6.2 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 126 s; peak memory 6.2 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Exact calculations for finite claims, cert-machine
Their verdict
“REFUSED”
Claim tested or proved
The exact code did not decide it; cert-machine records this as refused, which is no evidence against the claim.
What it covers
a finite component
Detail
decided in part (pre-registration amendment 5): the product inequality (b) is decided only on C x S, S x C and C' x C' (about 0.4% of S x S by volume; the rest projected above 20 CPU-hours at ~5 s per cell); the scalar certificate and inequality (a) on all of S x S are decided
Exact calculations for finite claims, cert-machine
Their verdict
“REFUSED”
Claim tested or proved
The exact code did not decide it; cert-machine records this as refused, which is no evidence against the claim.
What it covers
a finite component
Detail
decided in part (pre-registration amendment 5): the interval certificate is decided only for q <= 4/5; q > 4/5, which holds the near-uniform corner where P ~ 2e-5, is not (extending band 2 to q <= 9/10 alone took 84,610 boxes and 1,916 s); the pair certificate (all 14 polynomials) and the grid certificate (all 28 stopping iterations) are decided whole
Supercritical reconstruction on regular and Poisson trees
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
Three-state reconstruction above the Kesten–Stigum threshold
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.